dielectric properties of li o-3b o glasses · 2010. 7. 13. · dielectric properties of li2o-3b2o3...
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Dielectric properties of Li2O-3B2O3 glasses
Rahul Vaish and K. B. R. Varma*
Materials Research Centre, Indian Institute of Science, Bangalore-560 012,
India.
*Corresponding Author; E-Mail: [email protected];
FAX: 91-80-23600683; Tel. No: 91-80-22932914
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Abstract:
The frequency and temperature dependence of the dielectric constant and the electrical
conductivity of the transparent glasses in the composition Li2O-3B2O3 (LBO) were
investigated in the 100 Hz- 10 MHz frequency range. The dielectric constant and the loss
in the low frequency regime were electrode material dependent. Dielectric and electrical
relaxations were respectively analyzed using the Cole-Cole and electric modulus
formalisms. The dielectric relaxation mechanism was discussed in the framework of
electrode and charge carrier (hopping of the ions) related polarization using generalized
Cole-Cole expression. The frequency dependent electrical conductivity was rationalized
using Jonscher’s power law. The activation energy associated with the dc conductivity
was 0.80 ± 0.02 eV, which was ascribed to the motion of Li+ ions in the glass matrix. The
activation energy associated with dielectric relaxation was almost equal to that of the dc
conductivity, indicating that the same species took part in both the processes.
Temperature dependent behavior of the frequency exponent (n) suggested that the
correlated barrier hopping model was the most apposite to rationalize the electrical
transport phenomenon in Li2O-3B2O3 glasses. These glasses on heating at 933 K/10h
resulted in the known non-linear optical phase LiB3O5.
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1. Introduction:
Noncentrosymmetric borate-based compounds have been becoming increasingly
important, because of their symmetry dependent properties such as piezoelectric,
pyroelectric, ferroelectric and non-linear optical. Various borate-based single crystals,
including BaB2O4 [1], BiB3O6 [2], SrB4O7 [3], CsLiB6O10 [4], LiNaB4O7 [5] have been
investigated and reported to be promising from their physical properties view point.
These compounds have attained importance particularly for their use in non-linear optical
devices.
Amongst various borate-based materials, LiB3O5 single crystals have been
reported to be promising from their non-linear optical, pyroelectric and piezoelectric
properties viewpoint [6-8]. It has high non-linear optic coefficient, large optical damage
threshold, wide transmission window, low cost associated with fairly good chemical and
mechanical stability. LiB3O5 belongs to the orthorhombic crystal class associated with
Pna21 space group. Because of the combination of the promising physical properties
indicated above, LiB3O5 as a functional material in the Li2O-B2O3 binary system has
attracted the attention of several researchers around the globe. The same binary system
also yields Li2B4O7 compound which is technologically important owing to its interesting
surface acoustic wave (SAW), piezoelectric and pyroelectric properties [9,10].
Though, the optical properties of LiB3O5 have been studied in detail [11, 12], the
literature on the electric properties is limited [6]. Since LiB3O5 is polar, it deserves much
attention from its electrical transport properties point of view as these properties have
direct influence on its pyroelectric and piezoelectric characteristics. Glass-ceramic route
of fabricating transparent materials at finer scale which eventually on heat treatment
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yields the desired crystalline phase is an industrially important method. Therefore, to
begin with, glasses in the composition Li2O-3B2O3 (which on heating at appropriate
temperatures yielded crystalline LiB3O5 phase) have been investigated for their dielectric
and electrical conductivity properties over the range of temperatures and frequencies that
are normally of interest in the applications of these materials. The experimental data have
been modeled using Cole-Cole relation [13], Jonscher’s power law [14] and electric
modulus formalism. The details pertaining to these studies are reported in the following
sections.
2. Experimental:
Transparent glasses in the composition Li2O-3B2O3 (LBO) were fabricated via the
conventional melt-quenching technique. For this, Li2CO3 and H3BO3 were mixed and
melted in a platinum crucible at 1173 K for 30 min. The batch weight was 20 gm. Melts
were quenched by pouring on a steel plate and pressed with another plate to obtain 1-2
mm thick glass plates. These glasses were annealed at 673 K for 12 h. The amorphous
nature of the as-quenched samples was confirmed by X-ray powder diffraction (XRD,
Philips PW1050/37) using Cu Kα radiation. The glassy characteristics were established
by differential scanning calorimetry (DSC, Model: Diamond DSC, Perkin Elmer) studies.
The capacitance and dielectric loss (D) measurements on the as-quenched
(annealed) polished glass plates of 1 and 3 mm in thickness using various electrode
materials (Ag-paint, sputtered Au and thermally evaporated Al) were done using
impedance gain phase analyzer (HP 4194 A) in the 100 Hz-10 MHz frequency range with
a signal strength of 0.5 Vrms at various temperatures (300–525 K). Thin silver leads were
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bonded to the sample using silver epoxy. Based on these data the dielectric constants
were evaluated by taking the dimensions and electrode geometry of the sample into
account.
3. Results and Discussion:
The DSC trace that was obtained in the 600- 950 K temperature range for the as-
quenched glass plates is shown in Fig. 1. It exhibits the glass transition (endotherm, 770
K) and exotherms in the 835 -910 K temperature range associated with the crystallization.
The XRD pattern obtained for the as-quenched sample (Li2O-3B2O3 in molar ratio) that is
shown in Fig.2 (a) confirms its amorphous state. In order to ascertain the crystalline
phases that are evolving in the above temperature range, the as-quenched samples were
heated to 823 K/4 h, 873 K/6 h and 933 K/10 h. The X-ray powder diffraction pattern
obtained for the 823 K/4 h heat-treated samples is depicted in Fig. 2 (b). The Bragg peaks
that are encountered in this pattern could be indexed to Li2B4O7 (major phase) and
Li2B8O13 (minor phase) phases. The XRD pattern [Fig. 2 (c)] that was obtained for the
873 K/6 h heat-treated sample also revealed the presence of two major phases
corresponding to Li2B4O7 and Li2B8O13. In order to ascertain the thermal stability of these
phases (Li2B4O7 and Li2B8O13), the as-quenched glasses were heat-treated at 933 K/10 h
(beyond the intense exotherm in Fig. 1). The XRD pattern that was obtained at room
temperature for this sample is shown in Fig. 2 (d). It is interesting to note that all the
Bragg peaks in this pattern could be assigned to LiB3O5 phase (a=8.446(2) Ǻ, b=7.380(2)
Ǻ, and c= 5.147(2) Ǻ). The as-quenched glasses are likely to have diborate and
tetraborate structural units which favor crystallization of Li2B4O7 and Li2B8O13 at the
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initial stages of crystallization [15]. Based on the XRD studies, the scheme of the
crystallization of Li2O-3B2O3 glasses may be illustrated as follows;
Li2O-3B2O3 ⎯⎯⎯ →⎯ hK 6/873 0.5Li2B4O7 + 0.5Li2B8O13 ⎯⎯⎯ →⎯ hK 10/933 2LiB3O5 (1)
The variation of the dielectric constant ( 'rε ) with frequency (100 Hz – 10 MHz) of
measurement for 1 mm thick LBO glass-plates (with silver paint electrodes) at different
temperatures is shown in Fig. 3. At all the temperatures under investigation, 'rε decreases
with increase in frequency. The decrease is significant, especially at low frequencies,
which may be associated with the mobile ion polarization combined with electrode
polarization. The low-frequency dispersion of 'rε gradually increases with increase in
temperature due to an increase in the electrode polarization as well as the thermal
activation associated with Li+ ions in the LBO glasses. The electrode polarization is
significant at high temperatures (423 K-523 K) and masks the bulk response of the
glasses in the low frequency regime. When the temperature rises, the dielectric dispersion
shifts towards higher frequencies.
To begin with, an attempt was made to rationalize the dielectric relaxation in LBO
glasses by using the Cole-Cole equation [13]:
( ) αωτεε
εε −∞
∞+
−+= 1
*
1 is
r (2)
where sε is the static dielectric constant, ∞ε is a high frequency value of the dielectric
constant, ω (=2πf) is the angular frequency, τ is the dielectric relaxation time and α is a
measure of distribution of relaxation times with values ranging from 0 to 1. For an ideal
Debye relaxation, α = 0 and α > 0 indicates that the relaxation has a distribution of
relaxation times. After solving Eq. 2 for the dielectric constant, one obtains
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( ) ( ) ( )( ) ( ) ( ) αα
α
ωτπαωτπαωτεε
εε 221
1'
2sin21]2sin1[
−−
−∞
∞++
+−+= s
r (3)
The experimental data on the variation of 'rε with frequency could not be fitted perfectly
using Eq. 3 in the entire frequency range since the Cole-Cole equation predicts nearly
constant 'rε in the low frequency regime, which is not true in the present case. This is due
to the fact that the electrode/space charge polarization is dominant at low frequencies as
depicted in Fig. 3. The above observations necessitate the inclusion of the electrical
conductivity term in the Cole-Cole equation to rationalize the 'rε versus frequency
behavior of LBO glasses in the whole frequency range. After adding the term that
reflects the electrode/space charge polarization in the Eq. 3, one arrives at [16]
( ) ( ) ( )( ) ( ) ( ) s
o
sr ωε
σωτπαωτπαωτεε
εε αα
α2
221
1'
2sin21]2sin1[+
++
+−+= −−
−∞
∞ (4)
where s (0, 1) is a constant and σ2 is the conductivity, a contribution from the space
charges. Solid lines in Fig. 3 are the fitted curves (Goodness of fit (R2) >0.999) of the
experimental results (100 Hz-10 MHz) according to Eq. 4. The parameters that are
obtained from the best fit at various temperatures are presented in table I. In order to
further elucidate the dielectric relaxation in LBO glasses, it is important to estimate the
activation energy associated with the relaxation process. The activation energy involved
in the relaxation process of ions could be obtained from the temperature dependent
relaxation time (table I) as
( )kTE
o expττ = (5)
where E is the activation energy associated with the relaxation process, oτ is the pre-
exponential factor, k is the Boltzmann constant, and T is the absolute temperature. Fig. 4
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depicts the plot of ln (τ) versus 1000/T along with linear fit (solid line) to the above
equation (Eq. 5). The value that is obtained for E is 0.77 ± 0.03eV, which is ascribed to
the motion of Li+ ions [17] in the glass matrix.
The variation of the dielectric loss (D) with the frequency at various temperatures
is shown in Fig. 5. The loss decreases with increase in frequency at different temperatures
(313 K-523 K). However, it increases with increase in temperature, which is attributed to
the increase in electrical conductivity of the glasses. A relaxation peak at 150 Hz was
encountered when the measurements were done at 523 K. In order to understand the
effect of electrode materials used on the dielectric relaxation of LBO glasses in the low
frequency regime, different electrode materials (silver paint, sputtered gold and thermally
evaporated aluminium) were used. Fig. 6 (a & b) shows the dielectric constant and the
loss behavior at 523 K for various electrode materials. Significant difference in the
dielectric constants for different electrode materials was observed at low frequencies
[Fig. 6 (a)]. However, all the plots merge in the high frequency regime (above 10 kHz).
This electrode independent behavior at high frequencies (10 kHz-10 MHz) is attributed to
the intrinsic dielectric response of the glasses. Interestingly, clear relaxation peaks were
observed in the frequency dependent dielectric loss plots [Fig. 6 (b)]. The frequency
associated with the dielectric relaxation was found to vary with the electrode materials
used suggesting that the above relaxation is ascribed to electrode polarization. All the
plots overlap in the high frequency region akin to that of the dielectric constant plots [Fig.
6 (a)]. Inorder to probe further into these results the dielectric measurements were
performed on the samples of two different thicknesses (1 mm and 3 mm) using silver
paint electrodes at 523 K (Fig. 7). The dielectric dispersion is found to be dependent on
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the thickness of the sample indicating space charge polarization at sample/electrode
interfaces contributing to the observed dielectric dispersion.
Electric modulus formalism was also invoked to rationalize the dielectric response
of the present glasses. The use of electric modulus approach helps in understanding the
bulk response of moderately conducting samples. This would facilitate to circumvent the
problems caused by electrical conduction which might mask the dielectric relaxation
processes. The complex electric modulus (M*) is defined in terms of the complex
dielectric constant (ε*) and is represented as [18]:
M* = (ε*)-1 (6)
2"2'
"
2"2'
'*
)()()()("'
rr
r
rr
r iiMMMεε
εεε
ε+
++
=+= (7)
where 'M , "M and , 'rε , "
rε are the real and imaginary parts of the electric modulus and
dielectric constants, respectively. The real and imaginary parts of the modulus at different
temperatures are calculated using Eq. 7 for the LBO glasses and depicted in Figs. 8 (a &
b), respectively. One would notice from Fig. 8 (a) that at low frequencies, 'M approaches
zero at all the temperatures under study suggesting the suppression of the electrode
polarization. 'M reaches a maximum value corresponding to M∞ = ( ∞ε )-1 due to the
relaxation process. It is also observed that the value of M∞ decreases with the increase in
temperature. The imaginary part of the electric modulus (Fig. 8 (b)) is indicative of the
energy loss under electric field. The "M peak shifts to higher frequencies with increasing
temperature. This evidently suggests the involvement of temperature dependent
relaxation processes in the present glasses. The frequency region below the "M peak
indicates the range in which Li+ ions drift to long distances. In the frequency range which
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is above the peak, the ions are spatially confined to potential wells and free to move
within the wells. The frequency range where the peak occurs is suggestive of the
transition from long-range to short-range mobility. The electric modulus (M*) could be
expressed as the Fourier transform of a relaxation function φ(t):
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−−−= ∫
∞
∞0
* )exp(1 dtdtdtMM φω (8)
where the function φ(t) is the time evolution of the electric field within the materials and
is usually taken as the Kohlrausch-Williams-Watts (KWW) function [19,20]:
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞⎜
⎝⎛−=
β
τφm
tt exp)( (9)
where τm is the conductivity relaxation time and the exponent β (0 1] indicates the
deviation from Debye-type relaxation. The value of β could be determined by fitting the
experimental data in the above equations. But it is desirable to reduce the number of
adjustable parameters while fitting the experimental data. Keeping this point in view, the
electric modulus behavior of the present glass system is rationalized by invoking
modified KWW function suggested by Bergman. The imaginary part of the electric
modulus ( "M ) is defined as [21]:
( ) ( )[ ]βωωωωββ
ββ .1
)1("
"
MaxMax
MaxMM
++
+−= (10)
where "MaxM is the peak value of the "M and ωMax is the corresponding frequency. The
above equation (Eq. 10) could effectively be described for β ≥ 0.4. Theoretical fit of Eq.
10 to the experimental data is shown in Fig. 8 (b) as the solid lines. It is seen that the
experimental data are well fitted to this model except in the high frequency regime. From
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the fitting of "M versus frequency plots, the value of β was determined and found to be
temperature dependent. The plot of β versus temperature is depicted in Fig. 9. β
increases gradually with the increase in temperature indicating that as the temperature
increases the glass network loosens and the interactions between Li+ ions and
surrounding matrix decreases.
The relaxation frequency associated with this process was determined from the
plot of "M versus frequency. The activation energy involved in the relaxation process of
ions could be obtained from the temperature dependent frequency associated with the
peak of "M as:
⎟⎠⎞
⎜⎝⎛−=
kTEff R
om exp (11)
where ER is the activation energy associated with the relaxation process, fo is the pre-
exponential factor, k is the Boltzmann constant and T is the absolute temperature. Fig. 10
shows a plot between ln (fm) and 1000/T along with the theoretical fit (solid line) to the
above equation (Eq. 11). The value that is obtained for ER is 0.80 ± 0.02eV, which is
ascribed to the motion of Li+ ions and is consistent with the one reported in the literature
[17].
In order to elucidate the electrical transport mechanism in LBO glasses, DC
conductivity at different temperatures (σDC(T)), was calculated from the electric modulus
data. The DC conductivity could be obtained according to the expression [22]:
( )
( ) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎠⎞⎜
⎝⎛Γ
=∞
T
TTTM
Tm
oDC
β
βτ
εσ
1)(*)()( (12)
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where oε is the free space dielectric constant, M∞ (T) is the reciprocal of high frequency
dielectric constant and τm (T) (=1/2πfm) is the temperature dependent relaxation time. Fig.
11 shows the DC conductivity data obtained from the above expression (Eq. 12) at
various temperatures. The activation energy for the DC conductivity was calculated from
the plot of ln (σDC) versus 1000/T for LBO glasses, which is shown in Fig. 11. The plot is
found to be linear and fitted using the following Arrhenius equation,
⎟⎠⎞⎜
⎝⎛−= kT
EBT DCDC exp)(σ (13)
where B is the pre-exponential factor, EDC is the activation energy for the DC conduction.
The activation energy calculated from the slope of the fitted line is found to be 0.79 ±
0.03eV. This value of activation energy is higher than that of the value associated with dc
conduction in Li2O-2B2O3 glasses [23]. This is due to fact that in the alkali borate
systems, the local structure of boron could be tailored by varying the alkali oxide content.
At higher alkali content, more number of non-bridging oxygens (NBOs) are formed
which yield a open structure of the borate network [24]. Li2O-2B2O3 glasses have higher
molar content of Li2O than that of the Li2O-3B2O3 glasses which consequences the
change in the coordination of boron associated with the formation of NBOs. The
environment around Li+ is changed due to variation in the NBOs. Such structural changes
can have important influence on the mobility of Li+ ions. The Li+ mobility increases in
the presence of NBOs. This suggests that the glasses in the composition of Li2O-2B2O3
would have higher conductivity than that of the Li2O-3B2O3 glasses.
AC conductivity at different frequencies and temperatures, was determined by using the
dielectric data using the following formula:
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'roAC Dεωεσ = (14)
where ACσ is the AC conductivity at a frequency ω (=2πf). The frequency dependence of
the AC conductivity at different temperatures is shown in Fig. 12. At low frequency, the
conductivity shows a flat response which corresponds to the dc part of the conductivity.
At higher frequencies, the conductivity shows a dispersion. It is clear from the figure that
the flat region increases with the increase in temperature. The phenomenon of the
conductivity dispersion in solids is generally analyzed using Jonscher’s law
nDCAC Aωσσ += (15)
where DCσ is the dc conductivity, A is the temperature dependent constant and n is the
power law exponent which generally varies between 0 and 1. The exponent n represents
the degree of interaction between the mobile ions. The present glasses are found to obey
the above mentioned universal power law at all the temperatures and frequencies under
study. The theoretically fitted lines of Eq. 15 to the experimental data are shown in Fig.
12 (solid lines). The conductivity obtained for the present glasses at 500 Hz and 373 K is
1.5 X 10-7 Ω-1.m-1 which is in the same order of magnitude for LiB3O5 single crystals
along a and b-axes (9.5 X 10-8 Ω-1.m-1) [6]. However, slightly higher value of
conductivity associated with Li2O-3B2O3 glass is attributed to the easy migration of Li+
ions through the diborate and tetraborate structural units.
The variation of exponent n as a function of temperature is depicted in Fig. 9. It is
known that the conductivity mechanism in any material could be understood from the
temperature dependent behavior of n. To ascertain the electrical conduction mechanism
in the materials, various models have been proposed [25]. These models include quantum
mechanical tunneling model (QMT), the overlapping large-polaron tunneling model
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(OLPT) and the correlated barrier hopping model (CBH). According to the QMT model,
the value of exponent n is found to be 0.8 and increases slightly with increase in the
temperature whereas the OLPT model predicts the frequency and temperature
dependence of n. In the CBH model, the temperature dependent behavior of n is
proposed. This model states that the charge transport between localized states due to
hopping over the potential barriers and predicts a decrease in the value of n with the
increase in temperature, which is consistent with the behavior of n for the glasses
understudy (Fig. 9). This suggests that the conductivity behavior of LBO glasses can be
explained using correlated barrier hopping model.
The present glasses do not seem to follow the Ngai’s relation (β =1-n) [26] as the
plots of imaginary part of electric modulus are not fitted exactly in the high frequency
regime which influences the value of β [Fig. 8 (b)]. Since the values for β and n are
estimated in different frequency regions (as they could not be fitted well in the same
frequency region), it is inconsistent with the Ngai’s relation. Although the qualitative
changes in the values of β and n are in conformity with the fact that both parameters
represent the interaction between the ions [27].
The temperature dependence of the AC conductivity at different frequencies is
shown in Fig. 13. At high temperatures and low frequencies the curves tend to merge
with each other with a constant slope. This frequency independent behavior is attributed
to the contribution from the DC conduction. The solid line that is shown in Fig. 13 is the
linear fit. The slope of which gives the activation energy which is about 0.82 ± 0.03eV
attributed to the Li+ ion transport. It is worth noting that the activation energies for
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relaxation process and DC conduction are in close agreement. It suggests that similar
energy barriers are involved in both the relaxation and conduction processes.
4. Conclusions:
The frequency and temperature dependence of dielectric properties of Li2O-3B2O3
glasses were investigated in the frequency range of 100 Hz - 10 MHz. The dielectric
relaxation peak was observed in the frequency dependent dielectric loss plots whose
magnitude had electrode materials dependence. The dielectric relaxation behavior of
these glasses was rationalized using Cole-Cole equation and the electrical transport
properties were investigated and found to be obeying Jonscher’s universal law. The
activation energy associated with the dielectric relaxation determined from the dielectric
and electric modulus spectra was found to be 0.78 ± 0.04 eV, close to that the activation
energy for DC conductivity (0.80 ± 0.02 eV).
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Table I: Fitted parameters from the Cole-Cole equation for Li2O-3B2O3 glasses.
T (K)
sε ∞ε τ (µs)
α 2σ ( m..Ω )-1
s
423 19 8.5 80 0.428 7.8 E-10 0.39 448 19.8 8.8 20 0.401 2.31E-9 0.53 473 20.7 9 5.6 0.34 2.17E-7 0.98 498 21.5 9.4 2.63 0.336 5.85E-7 0.99 523 22 9.85 1.46 0.30 5.25E-6 0.99
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Figure captions:
Fig. 1: DSC trace for as-quenched Li2O-3B2O3 glass plates.
Fig. 2: X-ray powder diffraction patterns for the (a) as-quenched, (b) 823 K/4 h,
(c) 873 K/6 h and (d) 933 K/10 h heat-treated Li2O-3B2O3 glasses.
Fig. 3: Frequency dependent dielectric constant plots at various temperatures and solid
lines are the fitted curves using Eq. 3 in the text.
Fig. 4: ln (τ) versus 1000/T plot for Li2O-3B2O3 glasses.
Fig. 5: Dielectric loss versus frequency plots at various temperatures.
Fig. 6: Frequency dependent behavior of (a) Dielectric constant and (b) dielectric loss
using various electrode materials at 523 K.
Fig. 7: Variation in dielectric constant with frequency for the samples of two different
thicknesses (1 mm and 3 mm).
Fig. 8: (a) Real and (b) imaginary parts of the electric modulus as a function of
frequency at various temperatures. The solid lines are the theoretical fits.
Fig. 9: n & β versus T for Li2O-3B2O3 glasses.
Fig. 10: Arrhenius plot for electrical relaxation.
Fig. 11: Arrhenius plot for DC conductivity.
Fig. 12: Variation of AC conductivity as a function of frequency at different temperatures
and solid lines are the fitted curves.
Fig. 13: Temperature dependence of AC conductivity at different frequencies and solid
line is the linear fit.
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